Waddling Random Walk: Fast and Accurate Sampling of Motif Statistics in Large Graphs.

The relative frequency of small subgraphs within a large graph, such as one representing an online social network, is of high interest to sociologists, computer scientists and marketeers alike. However, the computation of these network motif statistics via naive enumeration is infeasible for either its prohibitive computational costs or access restrictions on the full graph data. Methods to estimate the motif statistics based on random walks by sampling only a small fraction of the subgraphs in the large graph address both of these challenges. In this paper, we present a new algorithm, called the Waddling Random Walk (WRW), which is significantly faster than the best of previously known algorithms. It derives its name from the fact that it sways a little to the left and to the right, thus also sampling nodes not directly on the path of the random walk. The WRW algorithm achieves its computational efficiency by not trying to enumerate subgraphs around the random walk but instead using a randomized protocol to sample subgraphs in the neighborhood of the nodes visited by the walk. In addition, WRW achieves significantly higher accuracy (measured by the closeness of its estimate to the correct value) and higher precision (measured by the low variance in its estimations) than the current state-of-the-art algorithms. We illustrate these advantages in speed, accuracy and precision using simulations on well-known and widely used graph datasets representing real networks.